Why are my seams becoming separate pieces in my UV editor?
Please use the terminology correctly. There are no separate seams in your UV editor. These are called UV islands. Also you do not apply a texture to faces, you can assign a material to faces and that material may use various textures for various properties of it. You can hover your mouse over some face, edge or vertex and press l to select linked elements limited by seams(ctrl+l to select linked with something already selected), you will need to make sure that Seam is selected under Delimt in the F6 panel or at the bottom of the t panel just after you press l same way as you can set properties for any other operation in Blender just after it is performed.



You did not describe what steps you take before the problem appears so it is hard to tell where you are making the mistake, but the process of UV mapping in Blender is quite simple, so if you follow it right, you should avoid any issues like this.


In edit mode select edges that you want as seams for the UV islands make sure they will be possible to unwrap to 2D islands and mark them as seams(ctrl+e -> a), there is also a clear seam option in the ctrl+e menu. 
Select all the geometry you wish to unwrap in edit mode(could be only some faces but in your case it is all the geometry so hit a to deselect whatever you have selected and then a again to select all) and unwrap the UVs(u -> u in 3d viewport). I think here is where you made the mistake - I am guessing you only selected some faces when unwrapping instead of all you wanted. 


That's it. Now if you are happy with the UVs you would assign a material to your mesh and then paint the textures for it. You can also pin UVs(p to pin alt+p to unpin) in the UV editor and then re-unwrap so Blender will keep the pinned UVs in place and will recalculate other UVs so they spread uniformly keeping the pinned UV in the same place. You will have to play around with it to see where this is useful.